TeamCity Projects and Multiple SVN Branches

You only need to create a single project, but you will need multiple build configurations - 1 for each branch. As far as I know, you can't customize the artifact folder name on disk (it's an auto-increment number), however you can download all artifacts as a zip file in TeamCity 4.5 from the UI. There is also a scheduler included with TeamCity that lets you cleanup artifacts so they don't consume too much disk space.

TeamCity 2018.1.5

TeamCity doesn't support multi branches for SVN as for GIT - so I solved such problem with Configuration parameter - where I set active branch from which I need to build and after can easily switch to another branch by running a custom build or change that configuration parameter.

After need just configures triggers to start building from a specific branch:

So on project side you can see different branches

And easily switch between branches by running Custom Build and change branch there: